Computation of confidence intervals in regression utilizing uncertain prior information

TL;DR

This paper develops a computationally feasible method for constructing confidence intervals in linear regression that incorporate uncertain prior information, improving efficiency when the prior is correct while remaining robust.

Contribution

It introduces a practical computational approach for a new confidence interval that leverages uncertain prior information in regression models.

Findings

Interval has smaller expected length when prior info is correct

Interval defaults to standard when data contradicts prior

Computational methods make the interval calculation feasible

Abstract

We consider a linear regression model with regression parameter beta =(beta_1, ..., beta_p) and independent and identically N(0, sigma^2)distributed errors. Suppose that the parameter of interest is theta = a^T beta where a is a specified vector. Define the parameter tau = c^T beta - t where the vector c and the number t are specified and a and c are linearly independent. Also suppose that we have uncertain prior information that tau = 0. Kabaila and Giri (2009c) present a new frequentist 1-alpha confidence interval for theta that utilizes this prior information. This interval has expected length that (a) is relatively small when the prior information about tau is correct and (b) has a maximum value that is not too large. It coincides with the standard 1-alpha confidence interval (obtained by fitting the full model to the data) when the data strongly contradicts the prior information. At first sight, the computation of this new confidence interval seems to be infeasible. However, by the use of the various computational devices that are presented in detail in the present paper, this computation becomes feasible and practicable.